Science:Math Exam Resources/Courses/MATH102/December 2014/Question C 01/Solution 1

Let $x=\cot(\theta )$, we can rewrite it as $x={\frac {\cos(\theta )}{\sin(\theta )}},$ With implicit differentiation we take the derivative of above expression with respect to $x$ on both sides to get

Graphs

${\begin{aligned}1&={\frac {-\sin ^{2}(\theta )-\cos ^{2}(\theta )}{\sin ^{2}(\theta )}}\cdot \theta ',\\1&=-{\frac {1}{\sin ^{2}(\theta )}}\cdot \theta '\end{aligned}}$

So we obtain $\theta '=-\sin ^{2}(\theta )=-{\frac {1}{1+x^{2}}},$ as shown in the following diagram

Note: the triangle drawn on the right diagram can be solely constructed from the equation of $x=\cot(\theta )$